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Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Northeast
2009 Regional Olympiad of Mexico Northeast
2009 Regional Olympiad of Mexico Northeast
Part of
Regional Olympiad of Mexico Northeast
Subcontests
(2)
1
1
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sequence of numbers defined by diagonal of a numbered square spiral
Consider the sequence
{
1
,
3
,
13
,
31
,
.
.
.
}
\{1,3,13,31,...\}
{
1
,
3
,
13
,
31
,
...
}
that is obtained by following diagonally the following array of numbers in a spiral. Find the number in the
100
100
100
th position of that sequence.https://cdn.artofproblemsolving.com/attachments/b/d/3531353472a748e3e0b1497a088472691f67fd.png
3
1
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NM tangent to incircle of ABC, 2 circumcircles related
The incircle of triangle
△
A
B
C
\vartriangle ABC
△
A
BC
is tangent to side
A
B
AB
A
B
at point
P
P
P
and to side
B
C
BC
BC
at point
Q
Q
Q
. The circle passing through points
A
,
P
,
Q
A,P,Q
A
,
P
,
Q
intersects line
B
C
BC
BC
a second time at
M
M
M
and the circle passes through the points
C
,
P
,
Q
C,P,Q
C
,
P
,
Q
and cuts the line
A
B
AB
A
B
a second time at point
N
N
N
. Prove that
N
M
NM
NM
is tangent to the incircle of
A
B
C
ABC
A
BC
.